All That Glitters is not Golden

Over many years of teaching, I realized that most students, and many teachers had extensive misunderstandings about the "Golden Mean" and it's history.

I want to try to dispel, and expand, on some of these common misunderstandings.  For example, many think that the "Golden Mean" was known to the early Greeks (it was) by that name (it wasn't).   The idea that Euclid labeled the idea, which was found in geometric constructions such as the pentagon (and pentagram), as "A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the lesser". The "extreme and mean ratio" is still frequently used to describe the idea.


Others believe it did not exist until Fibonacci created the Fibonacci numbers from which it was derived as a limit of the ratio of consecutive terms. First, Fibonacci did not create the, so called, Fibonacci sequence. It was known to the Indian Mathematicians as early as Pingalia before 200 BC. Fibonacci's Liber Abaci (1202) included both the means and extreme property, and the famous sequence, but it seems he never realized that the ratio of consecutive terms of the sequence would approach the well known ratio. Luca Pacioli gave the name "Divine Proportion" to his 1509 book about the ratio, illustrated by Leonardo da Vinci. Leonardo first used golden for the ratio by using the latin "secto aurea" (golden section)  The first use in English did not occur until mathematician James Sulley used it in 1875, according to Alfred Posamentior.  And it was "mathematician Simon Jacob (d. 1564) noted that consecutive Fibonacci numbers converge to the golden ratio;this was rediscovered by Johannes Kepler in 1608." *Wik





Perhaps the most common misconception of students is that the Golden ratio is some how a one-off.  A very special number with nothing else like it.  In truth, it is part of a class of numbers known as Pisot Numbers.  In fact, it is one of an infinite set of  numbers that are solutions to a quadratic equation, sharing many of the "special" qualities of the golden ratio.

The golden ratio is the smallest of these, but others include the "silver ratio" \( 1 + \sqrt{2} \) , and the "bronze" ratio, \( \frac{3+\sqrt(5)}{2} \).  The three numbers are roots  of the quadratic terms \(x^2 - x -1\), \(x^2-2x-1\) and \(x^2 - 3x - 1\). (There is a pattern here, it WILL be back)


I will try to point out how some of those "special" qualities of the golden mean are shared by these other metalic means.

One of the things that impress students is the continued fraction for the golden mean repeats the same number over and over:
\( \phi = 1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\dots}}}\)



but very few realize that there is a very similar expansion for the "silver mean",

\( \phi_2 = 2+\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{2+\dots}}}\)

and I bet most of them can figure out how to write the bronze (or third metallic) mean, and all the ones that come after it.


How about looking at a different way to write the positive roots of each of the metallic means I gave above.

The Goilden mean is \( \frac{1+\sqrt{4+(1^2)}}{2}\) ; The Silver mean is \( \frac{2+\sqrt{4+(2^2)}}{2}\); and the third metallic mean is \( \frac{3+\sqrt{4+(3^2)}}{2}\)... the ones following are equally evident.

Students confusion with history is somewhat confused by the fact that they are often introduced to the Golden mean in association with it's relationship with the Fibonacci sequence; 1, 1, 2, 3, 5, 8, ... and the ratio of consecutive terms approaches the Golden mean.
For the Silver mean, there is another well known(but not often to high school students) sequence that is mistakenly attributed to English mathematician John Pell. The sequences is 0, 1, 2, 5, 12, 29,... and bright students can quickly guess the Fibonacci-type recursive formula for this, and will probably anticipate that the sequence for the third metallic ratio would be 0, 1, 3, 10, 33,... and probably all the metallic ratios after that.

In Geometry students are familiar with the fact that the Golden mean can be found in the pentagon, between a diagonal and a side, or between the two sections of the intersection of diagonals.

The Silver mean is found in the ratio between a side and the second shortest diagonal

Unfortunately, that's where the sequence ends.  There are no regular polygons with ratios of sides and diagonals that are in the ratio of any other metallic mean.  As I will point out later, there are non-quadratic numbers that are Pisot numbers (or cubes and higher order) that I have not checked.

Some lesser known facts about the Metallic Means is that there powers approach "almost-integers" as higher (and not so much higher for many) powers.  For example \( \phi^7 \approx 29.03444\) and \( \phi^{13} \approx 529.0019\)  as you might expect from experience, odd powers overshoot the mark a smidge, evens undershoot.  The error diminishes logarithmetically.

IF we go to the other metallic ratios, they demonstrate the same behavior more quickly.  For example the silver mean  \( \phi_2^7 \approx 478.00209\) and \( \phi_2^{13} \approx 94642.000010\)  .

And the third metallic mean gives ( \( \phi_3^7 \approx 4287.00023) \)


Another interesting, and not well known fact about the Fibonacci sequence is that the digits Mod (n) have a repeat period.  For the Fibonacci period, they repeat their last digit, (mod (10) ) in a 60 digit cycle.

011235831459437 077415617853819 099875279651673 033695493257291

It turns out that this is true of all the metallic sequences, but it may be easier to spot in the shorter binary cycles.  The Fibonacci digits mod(3) cycle 0,1,1 repeatedly, (Even, Odd, Odd).   For the Pell sequence, the cycle is 0,1; and these two sequences alternate between the odd and even metal ratios.

 But base three is not too hard, so let's look at that cycle of remainders on division by three:  

For the Fibonacci sequence the cycle is 0, 1, 1, 2, 0, 2, 2, 1.

For the Pell Sequene the cycle is this cycle sort of the reverse of this, 0,1,2,2,0,2,1,1.  

And the third metallic sequence, cycles 0,1, similar to the binomial cycle.... (can you figure out why there is never a remainder of 2 when a bronze sequence is divided by 3?)  

After I first wrote this post, I came across a page called Goldennumber.net which had a nice compass rose illustration of the 60 cycle or the Fibonacci sequence in base ten. As noted, they credit a copyright to Lucian Khan.

Of interest is that the zeros occur equally spaced at the NESW compass points.  This is a pattern of many repeat cycles with metallic ratios, the zeros are equally spaced.  It is also easy to pick up from this that all the numbers at 30 degree multiples are fives, showing that F(5n) is divisible by five.  The page also pointed out that any two non -zero remainders that are 180 degrees apart on the wheel sum to ten. Students might want to explore similar patterns in wheels of  Fibonacci or other metallic sequences for remainders by other divisors. 

Other Pisot Numbers, including a Super-Golden Number, 

Wikopedia gives this description of the Pisot Numbers: 

In mathematics, a Pisot–Vijayaraghavan number, also called simply a Pisot number or a PV number, is a real algebraic integer greater than 1 all of whose Galois conjugates are less than 1 in absolute value. These numbers were discovered by Axel Thue in 1912 and rediscovered by G. H. Hardy in 1919 within the context of diophantine approximation. They became widely known after the publication of Charles Pisot's dissertation in 1938.

There are other sets that are roots of cubic (and higher order) equations, and the smallest possible Pisot Number is called the "Plastic Number" and is the real root of \(x^3-x-1\),

or approximately 1.324717957...


Just as the golden mean has it's value as the limit of the ratio of consecutive terms of the Fibonacci sequence, the plastic number can be derived from the Padovan sequence, first developed around 1994(?), P(0)=P(1)=P(2)=1; and P(n)=P(n-2) + P(n-1)  which begins, 1,1,1,2,2,3,4,5,7,9,12,16...

This sequence has a longer mod(2) cycle.  Like all Pisot numbers, they approach almost-integers, but they do so much slower than the powers of the Golden Mean.


There is even a Super-Golden Ratio which is the real root of \( x^3 -x^2 - 1\ )

or approximately 1.4655712318...

It has its related sequence also, Naryana's cows, which dates back to the 14th Century. Unlike Fibonacci's rabbits, the Cows go through three stages, immature, adolescent, and then mature, so only the matures reproduce. The pattern looks like The first few terms of the sequence are as follows: 1, 1, 1, 2, 3, 4, 6, 9, 13, 19,... . (students could have fun creating four or five stage maturation sequences and look for the limit of their ratios as a limit, and compare the qualities to those from these ratios.  )

As always, comments (and corrections) are welcomed.

The sequence on the far right is a variation of the Padovan Sequence which begins with